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%I #9 Feb 12 2023 01:46:49
%S 1,4,9,2,25,36,49,1,3,100,121,18,169,196,225,1,289,12,361,50,441,484,
%T 529,9,5,676,1,98,841,900,961,1,1089,1156,1225,6,1369,1444,1521,25,
%U 1681,1764,1849,242,75,2116,2209,9,7,20,2601,338,2809,4,3025,49,3249,3364
%N a(n) is the least number k such that k*n is a cubefull number (A036966).
%H Amiram Eldar, <a href="/A360541/b360541.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = 1 if and only if n is cubefull number (A036966).
%F a(n) = A356193(n)/n.
%F a(n) = A360539(n)^2/A329376(n)^3.
%F Multiplicative with a(p^e) = p^(max(e, 3) - e), i.e., a(p) = p^2, a(p^2) = p, and a(p^e) = 1 for e >= 3.
%F Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(2-s) - p^(-s) - p^(2-2*s) + p^(1-2*s) - p^(1-3*s) + p^(-3*s)).
%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(3)/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 2/p^5 - 1/p^6 - 1/p^8 + 2/p^9 - 1/p^10) = 0.2078815423... .
%t f[p_, e_] := p^(Max[e, 3] - e); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o (PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, f[i, 1]^(max(f[i, 2], 3) - f[i, 2]));}
%Y Cf. A036966, A329376, A356193, A360539.
%K nonn,easy,mult
%O 1,2
%A _Amiram Eldar_, Feb 11 2023
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